I'm a tribal member of the Cherokee Nation of Oklahoma and very familiar with the Park Hill Cemetery. First of all, is it possible this man was in unit in the Indian Territory/Oklahoma. The reason I ask is there are two (2) men listed in the Union Indian Home Guard units named John White. This would seem to explain the burial in the capitol of the Cherokee Nation. BTW, does he currently have a marker and you're wanting to provide a Civil War veteran's marker or is he in an unmarked grave?
Don't be confused by the reference below to Kansas Infantry. This is just the way these units are classified as they were generally formed in Kansas by Indian Territory refugees.
1st Regiment, Indian Home Guards, Kansas Infantry
3rd Regiment, Indian Home Guards, Kansas Infantry